What kind of ring is $U(L)$?
Since representations of Lie algebras behave like representations of groups (the category has tensor products and duals, for example), you should expect that the universal enveloping algebra $U(\mathfrak{g})$ has some extra structure which causes this, and it does: namely, it is a Hopf algebra (a structure shared by group algebras). The comultiplication is defined on basis elements $x \in \mathfrak{g}$ by
$$x \mapsto 1 \otimes x + x \otimes 1$$
(this is necessary for it to exponentiate to the usual comultiplication $g \mapsto g \otimes g$ on group algebras) and the antipode is defined by
$$x \mapsto -x$$
(again necessary to exponentiate to the usual antipode $g \mapsto g^{-1}$ on group algebras).
This is an important observation in the theory of quantum groups, among other things.
Thus, via the envelopping algebra Lie algebras and their represnetations cn be studied from a ring theoretic point of view. Is the cconverse true in some sense?
Not in the naive sense, the basic problem being that if $A$ is an algebra and $L(A)$ that same algebra regarded as a Lie algebra under the bracket $[a, b] = ab - ba$, then a representation of $L(A)$ does not in general extend to a representation of $A$, but to a representation of $U(L(A))$, which may be a very different algebra (take for example $A = \text{End}(\mathbb{C}^2)$).
Of course there are other relationships between ring theory and Lie theory. For example, if $A$ is a $k$-algebra then $\text{Der}_k(A)$, the space of $k$-linear derivations of $A$, naturally forms a Lie algebra under the commutator bracket. Roughly speaking this is the "Lie algebra of $\text{Aut}(A)$" in a way that is made precise for example in this blog post.
Best Answer
It depends. A rigorous treatment of Lie groups and Lie algebras (as in the style of, say, the excellent book Lie Groups: Beyond an Introduction by Knapp) does require a solid background in differential geometry and manifold theory. If the class is taught in this way, you may need to wait a year or two until after you have developed some expertise in these topics.
However, it is possible (and commonplace, these days) to provide an introduction to the subject 'via matrix groups' (as in the style of, say, Matrix groups for undergraduates by Tapp or Lie groups, Lie algebras and Representations by Brian C. Hall), where the topic is approached by beginning with the matrix groups we all know and love ($\operatorname{GL}_n, \operatorname{O}_n, \operatorname{SO}_n, \operatorname{U}_n, \operatorname{SU}_n$, etc.) and covering a lot of the theory using only these elementary groups. If the class is taught this way, you should be fine with the background you have.
At the end of the day, the best course of action is probably just to email the lecturer, and ask what approach they are taking, and what prerequisites they expect of their students.